Nonlinearity Computation for Sparse Boolean Functions

TL;DR

This paper introduces an algorithm to compute the nonlinearity of Boolean functions from their algebraic normal form, especially useful when traditional methods like the Fast Walsh transform are impractical for functions with more than 40 variables.

Contribution

It generalizes the weight expression in terms of ANF coefficients and formulates nonlinearity computation as a binary integer programming problem, enabling analysis of larger functions.

Findings

Effective for functions with over 40 variables

Reduces nonlinearity computation to binary integer programming

Provides an alternative when Fast Walsh transform is infeasible

Abstract

An algorithm for computing the nonlinearity of a Boolean function from its algebraic normal form (ANF) is proposed. By generalizing the expression of the weight of a Boolean function in terms of its ANF coefficients, a formulation of the distances to linear functions is obtained. The special structure of these distances can be exploited to reduce the task of nonlinearity computation to solving an associated binary integer programming problem. The proposed algorithm can be used in cases where applying the Fast Walsh transform is infeasible, typically when the number of input variables exceeds 40.